$\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}} = x$

Concept & Math When subtracting fractional radicals, extract the roots of the numerator and denominator separately where possible. Then, rationalize the denominators or find a common denominator to combine the terms. Step-by-Step Solution * Given expression: $\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}} = x$ * Apply the square root separately to the numerators and denominators: $\frac{\sqrt{4}}{\sqrt{3}} - \frac{\sqrt{3}}{\sqrt{4}}$ * Evaluate the perfect squares ($\sqrt{4} = 2$): $\frac{2}{\sqrt{3}} - \frac{\sqrt{3}}{2}$ * Find a common denominator to subtract the fractions. The common denominator is $2\sqrt{3}$. * Multiply the first term by $\frac{2}{2}$ and the second term by $\frac{\sqrt{3}}{\sqrt{3}}$: $\frac{2 \times 2}{2\sqrt{3}} - \frac{\sqrt{3} \times \sqrt{3}}{2\sqrt{3}}$ * Simplify the numerators: $\frac{4}{2\sqrt{3}} - \frac{3}{2\sqrt{3}}$ * Combine the fractions: $\frac{4 - 3}{2\sqrt{3}} = \frac{1}{2\sqrt{3}}$ * Thus, $x = \frac{1}{2\sqrt{3}}$. Exam Strategy & Shortcut You can treat $\sqrt{\frac{4}{3}}$ and $\sqrt{\frac{3}{4}}$ as variables $a$ and $\frac{1}{a}$ where $a = \frac{2}{\sqrt{3}}$. The expression is $a - \frac{1}{a}$. So, $\frac{2}{\sqrt{3}} - \frac{\sqrt{3}}{2}$. Cross-multiplying gives $\frac{4 - 3}{2\sqrt{3}} = \frac{1}{2\sqrt{3}}$. This visual shortcut skips redundant radical writing steps. Common Pitfall A common error is attempting to combine the terms under a single root by subtraction, incorrectly assuming $\sqrt{a} - \sqrt{b} = \sqrt{a - b}$. This leads to fundamentally flawed logic and wrong answers. Radicals must be evaluated or combined using common denominators. Final Answer Therefore, the correct answer is **$\frac{1}{2\sqrt{3}}$**.